Solving the modified complex short pulse equation of focusing type: a Riemann-Hilbert approach
In this paper, a Riemann-Hilbert approach is presented for the modified complex short pulse equation on the line with zero boundary conditions: \[ u_{xt}=u+\frac{1}{2}u^*(u^2)_{xx},\quad x\in\mathbb{R}, t>0, \] \[ u(x,0)=u_0(x),\quad x\in\mathbb{R} \] where \(u_0(x)\) is a smooth function which decays rapidly as \(x\to\pm\infty\). The Jost solutions of the associated Lax pairs are obtained and the results are used to formulate a matrix Riemann-Hilbert problem suitable for solving the Cauchy problem above. A parametric representation of the solution to the related Cauchy problem is obtained. Consequently, soliton solutions are constructed using the formalism of the Riemann-Hilbert problem.
scientific article; zbMATH DE number 7457156
scientific article